Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter1.1r

Text File | 1995-04-09 | 8KB | 249 lines

 247
àï1.1ïSolving Special Triangles

äïPlease find the indicated part of the following special
êêtriangles.
âêêïFind side "x" in the given triangle.

êêêïSince this is a 30°-60° triangle, side x
#êêêêè ┌─
#êêêïmust be 4á3 feet in length.

@fig1101.bmp,10,100
éS Both "special triangles" are right triangles and are referred
to as the 30°-60° triangle and the 45°-45° triangle.ïIf two sides, or
one side and one of the acute angles (< 90°) are known for one of ç
triangles, then you can predict the other sides and angles based on the
common proportions.ïThe reason for this is that all 30°-60° triangles
are similar, so their sides are proportional.ïThe same is true of
@fig1102.bmp,10,150
the 45°-45° triangle.

êêêLet's look at the 45°-45° triangle.ïSince two an-
êêêgles are equal, the two sides opposite ç angles
êêêare also equal in length.ïLet ç two sides have
êêêlength "a".ïWe can use the Pythagorean Theorem to
êêêfind the length of the hypotenuse.
#êêêêè (hyp)ìï=ïaì + aì
#êêêêêêï┌─
#êêêêë hypè=ïaá2
êêêThus, in every 45°-45° triangle, the sides
#êêêêêêï┌─
#êêêhave the proportions "aá2", "a", and "a"
êêêwith "a" opposite the 45° angle.
@fig1103.bmp,10,150

êêêè In the 30°-60° triangle, it is a little
êêêmore difficult to get the proportions.ïOne
êêêthing we know about a rectangle, however, is
êêêthat the diagonals are equal in length and
êêêbisect each other.ïIf in addition,
êêêangle A is 30° and angle B is 60° (see diagram), we
êêêcan determine more information.ïFor example, tri-
êêêangle ADC is isosceles making angle ACD equal to
êêê30°.ïThis in turn makes angle DCB equal to 60°,
êêêand it follows that angle BDC is also 60°.ïThus,
êêêtriangle BCD is equiangular, and hence equilateral.
êêêTherefore, side BC must have length one-half of the
êêêhypotenuse.ïWe now know that in every 30°-60°
@fig1104.bmp,10,118
êêêtriangle if the hypotenuse has length "c",
êêêthen the side opposite the 30° angle has
êêêlength "c/2". We can use the Pythagorean Theorem
êêêto find the length of the other leg.
#êêêêïcìï=ï(leg)ì + (c/2)ì
#êêêêï(leg)ìï=ïcì - (c/2)ì
#êêêêï(leg)ìï=ï3cì/4
#êêêêêè┌─────ëc ┌─
#êêêêïlegï=ïá3cì/4ï=ï─∙á3
êêêêêêê2
êêêThus, in every 30°-60° triangle, the sides have
#êêêêêêêêï┌─
@fig1104.bmp,10,118
#êêêthe proportions "c", "c/2", and "c/2∙á3" with
êêêc/2 opposite 30°.ï(Please see the Key Feature
êêêto understand more clearly the significance of
êêêç two triangles).
êêêè In the example, the hypotenuse has length 8
êêêfeet.ïThe side opposite the 30° angle must have
êêêone-half of this length or 4 feet, and
#êêêêêêêêêï┌─
#êêêthe side opposite the 60° angle has length 4á3
êêêfeet.
 1ïYou are given triangle ABC with angle C equal to 90°.ïIf
êè angle A equals 30° and side c equals 14, find side "a".

êêè A)ï14êêêïB)ï7
#êêêï┌─
#êêè C)ï7á3êêê D) å of ç
üêêïThe sides of a 30°-60° triangle have the
êêê proportions c for the hypotenuse, c/2 for the side
#êêêêêêêë┌─
#êêê opposite the 30° angle, and c/2∙á3 for the side
êêê opposite the 60° angle (See Details).ïSince this
êêê is a 30°-60° triangle with hypotenuse equal to
êêê 14, the side opposite the 30° angle must have one
êêê half this length or 7.
@fig1105.bmp,50,880
Ç B
 2ïYou are given triangle ABC with angle C equal to 90°.ïIf
êè angle B equals 60° and side "a" equals 5, find side "b".

êêè A)ï5êêêèB)ï10
#êêêï┌─
#êêè C)ï5á3êêê D) å of ç
üêêïThe sides of a 30°-60° triangle have the propor-
êêê tions c for the hypotenuse, c/2 for the side
#êêêêêêêë┌─
#êêê opposite the 30° angle, and c/2∙á3 for the side
êêê opposite the 60° angle (See Details).ïSince this
êêê is a 30°-60° triangle with the side opposite the
êêê 30° angle equal to 5, the side opposite the 60°
@fig1106.bmp,50,880
#êêêêêë┌─
#êêê angle must equal 5á3.
Ç C
 3ïYou are given triangle ABC with angle C equal to 90°.ïIf
#êêêêêêêï┌─
#êè angle B equals 45° and side "c" equals 4á2, find side "a".

êêè A)ï2êêêèB)ï4
#êêêï┌─
#êêè C)ï4á2êêê D) å of ç
üêêïThe sides of a 45°-45° triangle have the propor-
êêê tions "a" for the length of each leg, and
#êêêè ┌─
#êêê "a∙á2" for the length of the hypotenuse (See De-
êêê tails).ïSince this is a 45°-45° triangle with
#êêêêêê ┌─
#êêê hypotenuse equal to 4á2, the length of side "a"
êêê must be 4.
@fig1107.bmp,50,880
Ç B
 4ïYou are given triangle ABC with angle C equal to 90°.ïIf
#êêêê┌─
#êè side "c" equals 5á2, "a" = 5, and "b" = 5, find angle A.

êêè A)ï30°êêê B)ï60°

êêè C)ï45°êêê D) å of ç
üêêïIf the sides of a triangle have the proportions
#êêêè ┌─
#êêê "a∙á2", "a", and "a", then the triangle is a
êêê 45°-45° triangle.ïSince ç sides have those
êêê proportions, angle A must equal 45°.
@fig1108.bmp,50,880
Ç C
 5ïYou are given triangle ABC with angle C equal to 90°.ïIf
êè angle A equals π/3 and side "a" equals 24 feet, find
#êè side "b".êêêêè ┌─
#êêè A)ï8 ftêêêB)ï8á3 ft

êêè C)ï24 ftêêë D) å of ç
üêêïSince π/3 = (π/3)∙(180°/π) = 60°, this is a
@fig1109.bmp,50,880
êêê 30°-60° triangle.ïThe side opposite the 60°
#êêêêêë┌─êë ┌─
#êêê angle must be c/2∙á3.ïThus, c/2∙á3 = 24.
êêêê 48ê48√3
#êêê cï=ï───è=è────è=è16√3 feet
êêêê√3êï3
êêê The side opposite the 30° angle must be 8√3.
è Notice that angle A does not have to be the 30° angle.ïThe letters
used are arbitrary, whereas the location of the angle is the important
thing.
Ç B
 6ïYou are given triangle ABC with angle C equal to 90°.ïIf
êè side "a" equals 3 and side "b" equals 4, find side "c".

êêè A)ï5êêêèB)ï4
#êêêï┌─
#êêè C)ï3á2êêê D) å of ç
üêêïWe can use the Pythagorean Theorem to solve for
@fig1110.bmp,50,880
êêê side "c".
#êêêêë cìï=ï3ì + 4ì
#êêêêë cìï=ï25
êêêêêcï=ï5

êêêThis triangle is not as special as the 45°-45° or
the 30°-60° triangles, but it is interesting, because the sides have such
nice numbers.ïIt is called the 3-4-5 triangle.
Ç A
 7êêFind the value of "x" in the given figure.

#êêêêêêêê ┌─
#êêêïA)ï6êêêèB)ï6á2
#êêêê┌─
#êêêïC)ï8á6êêê D) å of ç
@fig1111.bmp,10,100
üïIn the 30°-60° triangle, 12 is opposite the 60° angle so it
must equal (c/2)∙√3.
#êêêêècè┌─
#êêêêè- ∙ á3ï=ï12
êêêêè2
#êêêêêêë┌─
#êêêê cï=ï24/√3è=ï8á3è ┌─
#Thus, the hypotenuse of the 30°-60° triangle is c = 8á3.ïSince this is
the leg of the 45°-45° triangle, it's hypotenuse, "x," must equal
(√2)∙(8√3)ï=ï8√6.
Ç C
 8êè Find the distance from A to D in the given figure.


êêêïA)ï90.6 ftêêè B)ï73.2 ft

êêêïC)ï80 ftêêë D) å of ç
@fig1112.bmp,10,100
üïSince triangle ABC is a 30°-60° triangle and the side opposite
the 30° angle is 100 feet, the side opposite the 60° angle, side AC,
must be 100√3 feet.ïSince triangle DBC is a 45°-45° triangle and one
leg is 100 feet, the other leg, side DC, must also equal 100 feet.ïThe
difference between ç two values is the answer.
#êêêêë┌─
#êêè AC - DCï=ï100á3 - 100è≈è73.2 feet

Ç B
 9êè Find the value of "x" in the given figure.


êêêïA)ï96.4 ftêêè B)ï136.6 ft

êêêïC)ï126.3 ftêêèD) å of ç
@fig1113.bmp,10,100
üïSince triangle BCD is a 45°-45° triangle, side DC is also of
length "x".ïSince triangle ABC is a 30°-60° triangle and sideïAC,
(100 + x), is opposite the 60° angle, (100 + x) must equal (c/2)∙√3.
êc
#Thus, ──ï=ï(100 + x)/√3, which is the length of the side opposite the
ê2
30° angle.ïMoreover, this expression also equals "x".

êêêë(100 + x)/√3ï=ïx
êêêê100 + xï= √3∙x
êêêë 100ï=ï(√3 - 1)∙x
êêêêxï≈ï136.6 feet
Ç B
 10êë Find the value of "x" in the given figure.


êêêïA)ï30êêêïB)ï25
#êêêê ┌─
#êêêïC)ï20á2êêêD) å of ç
@fig1114.bmp,25,118
üïFirst, π/3 = 60° and π/4 = 45°.ïSince side BC is opposite
the 60° angle, it has length 10√3.ïSince side BC is a leg of a 45°-
45° triangle, the hypotenuse, AB, must equal (10√3)∙(√2)ï=ï10√6.
Since side AB is opposite the 60° angle in triangle ADB, it must
equal (x/2)∙√3.
#êêêè x ┌─ê ┌─
#êêêè ─∙á3ï=ï10∙á6
êêêè 2
#êêêêë┌─
#êêêèxï=ï10∙á6∙2∙(1/√3)
#êêêêê ┌─
#êêêêxï=ï20á2
Ç C